Defines the URL protocols that MediaWiki will recognize as valid in wfParseUrl().
This is used in lots of places.
The most visible effect is it determines what protocols are allowed for external links: unrecognized protocols are ignored, no link is generated.
However, it's used in many other places too, including the code for Special:LinkSearch, and utility functions like wfParseUrl() that are used in a wide variety of different places.

Some examples that don't work (because "test://" is not defined as a valid protocol):

test://www.example.com

[test://www.example.com]

[test://www.example.com link text]

Some browsers, like Mozilla Firefox, will not follow file URLs on pages that have been loaded via HTTP. This is a security measure. See this mozillaZine article for more information, including some hints on how to overcome this restriction.

Most browsers do not support "remote file" URLs like file://host/share (only Internet Explorer handles these per default.
For Firefox, a plugin is required or use exactly 5 slashes like file://///host/share, see [1])

Warning:

In MediaWiki 1.20.2+, adding "file:" without trailing slashes to $wgUrlProtocols will break image embedding using the [[File:Image.jpg]] syntax.
This is a known bug related to a change that made URL protocol matching case-insensitive. As a workaround, use "file://" instead of "file:" as shown below.

To allow links to local files (which may make sense in an intranet context), put this into your LocalSettings.php:

$wgUrlProtocols=array('http://','https://','ftp://','ftps://',// If we allow ftp:// we should allow the secure version.'ssh://','sftp://',// SFTP > FTP'irc://','ircs://',// @bug 28503'xmpp:',// Another open communication protocol'sip:','sips:','gopher://','telnet://',// Well if we're going to support the above.. -ævar'nntp://',// @bug 3808 RFC 1738'worldwind://','mailto:','tel:',// If we can make emails linkable, why not phone numbers?'sms:',// Likewise this is standardized too'news:','svn://','git://','mms://','bitcoin:',// Even registerProtocolHandler whitelists this along with mailto:'magnet:',// No reason to reject torrents over magnet: when they're allowed over http://'urn:',// Allow URNs to be used in Microdata/RDFa <link ... href="urn:...">s'geo:',// urls define geo locations, they're useful in Microdata/RDFa and for coordinates'//',// for protocol-relative URLs);

MediaWiki versions:

1.18 – 1.21

$wgUrlProtocols=array('http://','https://','ftp://','irc://','ircs://',// @bug 28503'gopher://','telnet://',// Well if we're going to support the above.. -ævar'nntp://',// @bug 3808 RFC 1738'worldwind://','mailto:','news:','svn://','git://','mms://','//',// for protocol-relative URLs);

The default protocols should all be safe to click on (no evil side effects), and removing a protocol from the list will cause URLs using those protocols to become unrecognized in many places throughout the software.
In particular, removing 'http://' or other common protocols will probably break huge amounts of stuff.
Nevertheless, if you need to do so (for example, you already have a News: namespace), you can do something like this: